Unit 5, Lesson 8
Multiplying Rational Numbers (Part 1)
Grade 7
Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
None
This activity reminds students of previous work they have done with constant speed situations, using
Students may choose to use strategies such as creating a double number line or table of equivalent ratios to make sense of these problems and come up with a solution. While students are free to use these strategies, ensure that they also understand how to use
Ask students what they remember about problems involving distance, rate, and time. They might offer that distances traveled and elapsed time create a set of equivalent ratios or that the elapsed time can be multiplied by the speed to give the distance traveled. Give students 1 minute of quiet work time, and follow with a whole-class discussion.
A car is traveling at a constant speed of 60 miles per hour. How far does the car travel in:
The purpose of this discussion is to remind students of how the equation
Consider drawing a diagram or table to facilitate the discussion of each problem and to remind students of the strategies they used while working with proportional relationships, such as using a scale factor or calculating the constant of proportionality. When relating distance and time in a constant speed situation, the speed is the constant of proportionality.
MLR6: Three Reads
None
The purpose of this activity is for students to encounter a concrete situation where multiplying two positive numbers results in a positive number and multiplying a negative and a positive number results in a negative number.
Students use their earlier understanding of a chosen zero point, a location relative to this as a positive or negative quantity, and a description of movement left or right along the number line as negative or positive. They extend their understanding to movement with positive and negative velocities and different times. This situation will produce negative or positive end points depending on whether the velocity is negative or positive. Looking at a number of different examples will help students generalize about the sign of the product of a negative number and a positive number (MP8).
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the questions.
Give students 6–7 minutes of quiet work time, and follow with a whole-class discussion.
A traffic safety engineer was studying traffic patterns. She set up a camera to record the speed and direction of cars and trucks that passed by. She decided to represent positions to the east of the camera with positive numbers and positions to the west of the camera with negative numbers.
Number line. To the left is the word west. To the right is the word east. The numbers negative 50 through 50, in increments of 10, are indicated. There are tick marks midway between.
A car is traveling east at 12 meters per second. Where will it be 10 seconds after it passes the camera?
A car is traveling west at -14 meters per second. Where will it be 10 seconds after it passes the camera?
| velocity (meters per second) |
time after passing the camera (seconds) |
position (meters) |
equation | |
|---|---|---|---|---|
| car A | +12 | +10 | +120 | |
| car B | -14 | +10 | ||
| car C | +9 | +5 | ||
| car D | -11 | +8 | ||
| car E | -15 | +20 | ||
| car F | +8 | 0 |
Complete the sentences. Be prepared to explain your reasoning.
A positive number times a positive number equals a _______________________.
A negative number times a positive number equals a _______________________.
The purpose of this discussion is to emphasize that the product of two positive numbers is a positive number and that the product of a positive number and a negative number is a negative number. Begin by displaying the table and the number line from the Task Statement for all to see. Invite students to share their responses and reasoning for each car.
Demonstrate how cars with a positive velocity are moving towards the east and cars with a negative velocity are moving towards the west. Then place a point on the number line to represent each car’s position (except Car E’s position) at the given time. Discuss the following questions:
We can use signed numbers to represent the position of an object along a line. We pick a point to be the reference point and call it zero. Positions to the right of zero are positive. Positions to the left of zero are negative.
A number line with the numbers negative 10 through 10 indicated. A point is indicated at zero and is labeled "reference point." Another point is indicated at negative 4 and is labeled "4 units to the left of zero." A third point is indicated at 7 and is labeled "7 units to the right of zero."
When we combine speed with direction indicated by the sign of the number, it is called velocity. For example, if you are moving 5 meters per second to the right, then your velocity is +5 meters per second. If you are moving 5 meters per second to the left, then your velocity is -5 meters per second.
If you start at zero and move 5 meters per second for 10 seconds, you will be 50 meters to the right of zero. In other words,
If you start at zero and move -5 meters per second for 10 seconds, you will be 50 meters to the left of zero. In other words,
In general, a negative number times a positive number is a negative number.
Help us improve by sharing suggestions or reporting issues.
Encourage students who get stuck to use the provided number line to represent each situation.
The table shows how much carbon dioxide, on average, is released by each of these things in one year.
| object | average amount of carbon dioxide released in a year (kilograms) |
|---|---|
| campfire | 9 |
| car | 4,500 |
| semi-truck | 200,000 |
| oak tree | -22 |
| pine tree | -10 |
| oyster | -0.004 |
| clam | -0.003 |
Building Toward
Building Toward